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Understanding the geometry of curve graphs is important for proving results on mapping class groups of surfaces. In this talk, we will shed light on the geometry of curve graphs by describing “filtrations” of them by hyperbolic graphs. These graphs are arranged in a sequence via distance non-increasing maps, and the fibers are quasi-trees. This yields a new proof of finite asymptotic dimension of curve graphs. We also describe some useful aspects of the dynamics of the mapping class group actions on the graphs in the filtrations.